416 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1953 



CASES IN THE THREE-STAGE SWITCHING ARRAY WHERE N = r(MOD n) 



Table I indicated that for AT = 25 and n = 5 a total of 675 cross- 

 points were required. A square array requires only 625. Fig. 7 shows a 

 layout of switches where N = 25 and n = 3. In this case one input is 

 left over when 25 inputs are divided into threes. The lone input requires 

 three paths to the intermediary switches. This is in accordance with 

 Fig. 3. The lone output also requires three paths to the intermediary 

 switches. Also from Fig. 3, the lone input to the lone output requires 

 only one path. Hence there must be one switch capable of connecting 

 the lone input to the lone output. The number of crosspoints required 

 is 615 which is less than the 625 required by the square array. This 

 scheme can be extended to any case where N = kn -\- r, where the re- 

 mainder, r, is an integer greater than zero but less than n. The formula 

 for the number of crosspoints where k input and k output switches of 

 size n and one input and output switch of size r are used is: 



C = 2(2n - 1)(N - r) -I- 2(n + r - l)r + (n - r) 



+ {n + r- 1) i^-^ + lY - n + i 



(16) 



I. G. Wilson has pointed out that for a lone input the crosspoints in 

 the intermediary switches can be used to isolate its possible connections 

 hence no crosspoints are required in the input stage. This likewise ap- 

 plies for a lone output. With this modification the array in Fig. 7 requires 

 six fewer crosspoints. For this case, when r = 1, the number of cross- 

 points is: 



C = 2(2n - \){N - 1) + (n - 1) (^^^^^Y 



Table V — Minimum Values of A^ for Given Values of n 



